First-passage and first-hitting times of Levy flights and Levy walks

TL;DR

This paper analyzes the full distribution of first-passage and first-hitting times for Levy flights and Levy walks, highlighting differences caused by leapovers and the Levy stable index, with applications in search modeling and algorithms.

Contribution

It provides a comprehensive analysis of first-passage and first-hitting time distributions for Levy processes, emphasizing the effects of leapovers and the stable index.

Findings

Different behaviors for finite and infinite moments of jump lengths.

Distinct distributions for first-passage and first-hitting times due to leapovers.

Insights into short and long time limits of Levy processes.

Abstract

For both Levy flight and Levy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For Levy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ("leapovers"), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the Levy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms.